Yes, when the band hits prongs instead of the doughnut/base, then you need a spacer to be sure the diamonds in the band are not cutting into the prongs. I can't see either ring close up enough to tell if they do or not, but it probably is safest not to let them rub.
You need the opinion of a professional, which might be tough to get if they want to sell you that spacer... I would go to a trusted jeweler and ask their opinion - or maybe a couple of different jewelers. I don't think that any of us could tell from some images if you need a spacer or not. If you'd rather play it safe, and a spacer doesn't bother you, then add one for the peace of mind.